We consider the problem of implementing a fixed social choice function between multiple players (which takes as input a type ti from each player i and outputs an outcome f(t1,..., tn)), in which each player must be incentivized to follow the protocol. In particular, we study the communication requirements of a protocol which: (a) implements f, (b) implements f and computes payments that make it ex-post incentive compatible (EPIC) to follow the protocol, and (c) implements f and computes payments in a way that makes it dominant-strategy incentive compatible (DSIC) to follow the protocol. We show exponential separations between all three of these quantities, already for just two players. That is, we first construct an f such that f can be implemented in communication c, but any EPIC implementation of f (with any choice of payments) requires communication exp(c). This answers an open question of [Fadel and Segal, 2009; Babaioff et. al., 2013]. Second, we construct an f such that an EPIC protocol implements f with communication C, but all DSIC implementations of f require communication exp(C).